Galois theory allows one to reduce certain problems in field theory, especially those related to field extensions, to problems in group theory. For questions about field theory and not Galois theory, use the (field-theory) tag instead. For questions about abstractions of Galois theory, use (galois-...

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Galois theory reference request

I ran into the following description of Galois theory in Gelfand and Manin's Methods of Homological Algebra. But this looks like nothing like the Galois theory I know that deals with subgroups and ...
2
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1answer
123 views

Galois group of $x^{15}-1$

Let $\zeta$, $\eta$, $\omega$ denote the primitive fifteenth, fifth, and cube roots of unity. a) Describe all the automorphisms in $G=G(\mathbb Q (\zeta)/ \mathbb Q)$. b) Show that $G$ is isomorphic ...
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2answers
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Find the field of intersection

Let $\mathbb{F}_{p^{m}}$ and $\mathbb{F}_{p^{n}}$ be subfields of $\overline{Z}_p$ with $p^{m}$ and $p^{n}$ elements respectively. Find the field $\mathbb{F}_{p^{m}} \cap \mathbb{F}_{p^{n}}$. Could ...
2
votes
1answer
129 views

Proving irreducibility of polynomial over various fields

I am working on a problem set from an online course and am struggling to understand a key concept related to proving reducibility / split-ability etc. in a finite field $\mathbb{F}_{p^n}$, as well as ...
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1answer
49 views

Field algebraically closed

The problem is Let $E$ a finite extension of $F$ and $E$ is algebraically closed, show that $F$ is perfect. I know that a field $F$ is called perfect if every irreducible polynomial in $F[x]$ is ...
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0answers
53 views

Finding the Galois Correspondence of polynomial $t^4-2$

For the polynomial $t^4-2$ in $\Bbb Q[t]$, the splitting field is given by $\Bbb Q(\alpha, i)$ where $\alpha$ is $2^{1/4}$. I figured out that the Galois group of this polynomial is the dihedral ...
3
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2answers
198 views

Solvability of Artin-Schreier Polynomial

I'm having a hard time trying to prove that the polynomial f(x) = x^p - x - 1 in Z_p[x] is not solvable by radicals even though its Galois Group is solvable. So far, I have shown that the ...
4
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1answer
99 views

Galois groups of quintics

I'm trying to determine which subgroups of $S_5$ occur as the Galois group of an irreducible quintic $f\in\Bbb{Z}[X]$. I know such a subgroup of $S_5$ should be transitive, leaving only five ...
2
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1answer
67 views

Finite separable field extensions such that $KL/L$ and $K/K\cap L$ have non-isomorphic automorphism groups

If $K/F$, $L/F$ are finite separable extensions (not necessarily finite Galois extensions), then it seems clear that $KL/F$ is also a finite separable extension. However, in this case, is $\...
2
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1answer
40 views

Galois Group of a Product

Q: Let K be the splitting field for $(x^{5}-1)(x^{3}-2)$ over $\mathbb{Q}$. Compute the cardinality of the Galois group $G$ for $\mathbb{Q} \subset K$, and show that G is not abelian. So first I ...
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Galois group solvable but $f$ not solvable.

I know from a theorem that: Let $F$ be a field of characteristic $0$ and $f(x)\in F[x]$. Then $f(x)$ is solvable by radicals if and only if the Galois group of $f(x)$ is solvable. But what if the ...
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0answers
56 views

Proof of Version of Krasner's Lemma

I need some help in constructing the proof to this version of Krasner's Lemma from Serre's Local fields text book: Let E/K be a finite Galois extension of a complete field K. Prolong the valuation ...
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0answers
41 views

Polynomials in the extension of a complete field

I need some help in answering this exercise from Serre's Local Fields textbook: Let K be a complete field, and let f(X) in K[X] be a separable irreducible polynomial of degree n. Let L/K be the ...
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0answers
29 views

Is this a valid way to extend the proof of the insolubility of the quintic?

I'm just musing here, this is only (barely even) a half-formed idea, but I'm just wondering if it's at all a valid train of thought. The proof I read of the insolubility of the quintic polynomial ...
2
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0answers
113 views

Alternating groups as galois groups

Is there an elementary proof that the alternating group $A_n$, for any $n$, is the Galois group of an extension of the rationals? In fact, I am looking for a proof which does not use Hilbert's ...
2
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1answer
66 views

Which Galois Field is isomorphic to this extension?

Let $\alpha$ be an element in an algebraic closure of $GF(64)$ such that $\alpha^4=\alpha+1$. For which $r\in \mathbb{N}$ is $GF(64)$ adjoined $\alpha$ isomorphic to $GF(2^r)$? [Adding the following ...
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1answer
45 views

Reducing a sum of products of primitive nth roots

QUESTION: Suppose $\zeta$ is an primitive n-th root of unity. And suppose $n=p^r$ where $r\geq 1$. What is $(1-\zeta)\zeta^{p^r-p^{r-1}-1}$ written as the sum of the basis elements $1,\zeta,\zeta^2,......
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0answers
46 views

Finding polynomial with Galois group $S_n$.

I'm studying the proof that for every $n\in \mathbb{N}$, there exists a polynomial $f\in \mathbb{Q}$ such that $\mbox{Gal}(E/\mathbb{Q})\cong S_n$, with $E$ the splitting field of $f$ over $\mathbb{Q}$...
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0answers
53 views

Unramified Galois extension of a function field of a curve - definition

Reading some papers I've encountered the following: Let $X$ be a smooth curve over a field $K$ with function field $K(X)$. Consider the maximal Galois extension of $K(X)$ which is unramified over $X$. ...
0
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1answer
36 views

Calculate the degree of $[\mathbb Q(a):\mathbb Q]$

How to determine $[\mathbb Q(a):\mathbb Q]$ for $a\in\overline{\mathbb Q}\setminus\mathbb Q$ with $a^p=1$ with $p$ prime? So $a$ is an algebraic, complex number which cannot be written as a quotient. ...
4
votes
1answer
60 views

How can you write $\mathbb{Q}[\sqrt[3]{2},\omega_3]$ using a single algebraic element $\mathbb{Q}[\alpha]$?

Looking at the basis of $\mathbb{Q}[\sqrt[3]{2},\omega_3]$ gives me no idea on how to generate it using $\{1, \alpha, \alpha^2,\alpha^3,\alpha^4,\alpha^5\}$ for some $\alpha$ algebraic over $\mathbb{...
0
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1answer
55 views

How can I compute the galois group $\text{Gal}(K[\sqrt[n] X]/K)$?

1) How can I compute the galois group $\text{Gal}(K[\sqrt[n] X]/K)$ where $K=\mathbb C(X)$ ? I would say that the minimal polynomial of $\sqrt[n]X$ on $K$ is $t^n-X\in K[t]$ which is separable, ...
3
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1answer
63 views

Calculating the degree of $[\mathbb Q(\sqrt[n]{m}):\mathbb Q]$

Let $m\in\mathbb Z$ with a prime factorization of the form $m=p\Pi p_i^{n_i}$, $p\neq p_i$. How can I calculate $[\mathbb Q(\sqrt[n]{m}):\mathbb Q]$ for a natural number $n$?
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Compute Galois group of these extensions.

I think that I have a problem in the redaction. I have to compute 1) $G=\text{Gal}(\mathbb Q(\sqrt 3,\sqrt 2)/\mathbb Q)$ I know that $\mathbb Q(\sqrt 3,\sqrt 2)$ is the splitting field of $X^4-...
2
votes
1answer
152 views

Galois field extension and number of intermediate fields

Given the galois extension $L/K$, where $L=\mathbb{Q}(\zeta, \sqrt[3]{2})$. I've already calculated that $[L:\mathbb{Q}]=6$. I'm trying to prove that the number of intermediate fields is also 6, ...
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0answers
62 views

Calculating $Aut(\mathbb{Q}(\sqrt[3]{2}/\mathbb{Q}(\zeta))$

I am considering the Galois extension $L/\mathbb{Q}$, where $L = \mathbb{Q}(\sqrt[3]{2}, \zeta)$, and $\zeta$ is a primitive cube root of unity. I've found that $Aut(L/\mathbb{Q})$ can be viewed as ...
5
votes
0answers
82 views

Let $K = \mathbb{Z_p(t)}$ be the field of rational functions and let $f(x) = x^p - x - t \in K[x]$ and let $E/K$ be the splitting field of $f(x)$.

Show $f(x)$ is not solvable by radicals and $Gal(E/K) \cong Z_p$. I was just reading Galois theory from the textbook "Galois Theory - Rotman" and I stumbled acrossed an exercise that looked ...
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1answer
75 views

Intermediate field extensions and Degree of field extension

I was wondering whether there is a relation between $[L:K]$ and the number of intermediate fields $F$, where $K\subseteq F\subseteq L$. If there is then can someone please explain why. What if $L/K$ ...
2
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1answer
66 views

Compute degree and Galois group of a Galois extension

I'm trying to show that $[L:\mathbb{Q}]=8$, where $L=\mathbb{Q}(i, \sqrt{2}, \sqrt{3})$. I tried using the tower law to show this by saying: $[L:\mathbb{Q}]=[L:\mathbb{Q(\sqrt{2}, \sqrt{3})}]*[\...
5
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2answers
153 views

Finding a fixed subfield of $\mathbb{Q}(t)$

I'm looking at automorphisms $t \to 1-t$ and $t \to \frac{1}{t}$ of the field $\mathbb{Q}(t)$. By looking at the relations between these I think I've found the group generated by them to be $S_3$. ...
1
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1answer
75 views

Compositum of all Galois extensions of prime power degree

Let $\mathbb{K}$ a field of characteristic $\not =p$ prime, and $\mathbb{K}_p$ the compositum of all the Galois extensions of $\mathbb{K}$ whose degree is a power of $p$. I have to prove that for ...
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1answer
36 views

Question on Galois

1) If $E/F$ is a Galois extension and $B$ is an intermediate field, then $E/B$ is a Galois extension. If $E/F$ is a Galois extension, then $E$ is a splitting field of some $f(x)\in F[X]$. Can I just ...
1
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1answer
150 views

Basis for field extension $\mathbb{Q}(\zeta , \sqrt[3]{2})/\mathbb{Q}$

I'm trying to find a basis for the field extension $\mathbb{Q}(\zeta , \sqrt[3]{2})/\mathbb{Q}$, where $\zeta$ is the cube root of unity. I attempted this with starting with a set of elements I know ...
2
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1answer
76 views

Galois group of $f = x^5 - 2x^3 - x^2 + 2$

I've made some progress finding the Galois group of $f = x^5 - 2x^3 - x^2 + 2$ but I am having some difficulties. I've factorised it over $\mathbb{Q}$ as $f = (x^2 - 2)(x-1)(x^2 + x + 1)$ and so I ...
0
votes
1answer
62 views

Finding whether a field extension is normal and/or separable

I have the field extension $\frac{\mathbb{Q}(\sqrt[4]{-7})}{\mathbb{Q}(\sqrt{-7})}$ And i have to show whether it is normal and whether it is separable. I know that this extension is both separable ...
0
votes
1answer
197 views

Galois group of a polynomial and subfields

Suppose $f(x)\in \mathbb{Z}[x]$ is an irreducible quartic whose splitting field has Galois group $S_4$ over $\mathbb{Q}$. Let $\theta$ be a root of $f(x)$ and set $K=\mathbb{Q}(\theta)$. a) Prove ...
3
votes
2answers
175 views

Showing this field extension is not normal

This question is part of a homework assignment. We are asked if the field extension $\mathbb{Q(\sqrt[4]{-7})}/\mathbb{Q}$ is normal. Here is what I have so far: The obvious thing to do seems to be ...
1
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1answer
38 views

Given two dot products with the same vector in a prime finite field of 2 (Galois Field), how can one figure out future dot products?

I've stumbled upon an interesting "rule" derivation for the value of a dot product in $\mathbb{R}^{n}$ like this: Given an arbitrary vector $\vec a \in \mathbb{R}^{n}$ and the values of two dot ...
5
votes
1answer
59 views

Converse of Galois Theorem.

If $F\subset K\subset E$ field extension such that $K\subset E$ and $F\subset K$ is both finite and Galois extensions then $F\subset E$ is Galois extension. My intuition says that this is false but I ...
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0answers
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Let $f = x^4 + ax^2 + bx + c \in K[x]$ be an irreducible and separable polynomial and $L/K$ be its splitting field extension.

Let $disc(f) = D = \Delta^2$ be denoted as the discriminant of $f$ and assume that $\Delta \notin K.$ Let $g \in K[x]$ be the resolvent cubic of $f$ and assume there exists only one root which is $d \...
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0answers
45 views

question on Galois group

Let $f(x)\in F[x]$, let $E/F$ be a splitting field, and let $G=\text{Gal}(E/F)$ be the Galois group. 1) If $f(x)$ is irreducible, then $G$ acts transitively on the set of all roots of $f(x)$ (if $\...
2
votes
1answer
37 views

$F/E$ a finite Galois extension, the integral closure of $E[X]$ in $F(X)$

If $F/E$ is a finite Galois extension, then one can show that $F(X)/E(X)$ is also a finite Galois extension of the same degree (a basis for $F/E$ is also a basis for $F(X)/E(X)$). Since $E[X]$ is a ...
5
votes
0answers
64 views

Splitting field of $1 + x^3 + x^6 + x^9$

I'm studying for a qualifying exam and wanted to know if my reasoning on this problem was correct: Let $L$ be the splitting field of $f(x) = 1 + x^3 + x^6 + x^9$ over $\mathbb{Q}$. Find the Galois ...
1
vote
1answer
113 views

On Galois closure

I'm working on this problem in Hungerford: For $\sigma \in Aut_F \bar{F}$, show that every finite extension of $K$, the fixed field of $\sigma$, is cyclic. For a finite extension $L$ of $K$, let $M$ ...
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1answer
39 views

degree of extension of rationals adjoin infinitely many roots

Consider a radical extension of $\mathbb{Q}$ in which infinitely many numbers are adjoined. Each of these numbers is the $n^{th}$ root of some integer, where $n$ can vary. Can any information be ...
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0answers
55 views

Galois group of $E/K$ & Galois group of the extension $E$ over Fixed field?

I've proved this result for myself, but I have doubt in my proof whether it is true : Let $E/K$ be a field extension and $G(E/K)$ its Galois group. Suppose $E^{G(E/K)}$ is its Fixed field, i.e. $\{...
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0answers
63 views

Construct degree $n$ field extension with no intermediate field

I want to construct degree $n$ field extension with no intermediate field for each $n$. I know for any finite group $G$ there is a Galois extension $K/F$ so that $Gal(K/F)$ is $G$. So my idea was to ...
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votes
1answer
55 views

Fixed fields,cyclic groups and Galois theory

Let $G=S \times T$ where $S$,$T$ are both finite cyclic groups. Question 1: Is it true that there exists a Galois finite extension $L/F$ such that $Gal(L/F) \cong S \times T$? (I can't recall if ...
3
votes
1answer
112 views

$\{1,\sqrt [3]{7},{\sqrt [3]{7 }}^{2}\}$ is a basis for $Q[\sqrt {2},\sqrt {3},\sqrt [3]{7}]$ over $Q[\sqrt {2},\sqrt{3}]$

I am trying to prove that $\{ 1,\sqrt [ 3 ]{ 7 } ,{ \sqrt [ 3 ]{ 7 } }^{ 2 }\} \quad is\quad a\quad basis\quad for\quad Q[\sqrt { 2 } ,\sqrt { 3 } ,\sqrt [ 3 ]{ 7 } ]\quad over\quad Q[\sqrt { 2 } ,\...
0
votes
1answer
32 views

Splitting field of $x^ {a^n}$ −1 in Z/aZ[x]

What is the splitting field of the polynomial $x^ {a^n}$ −1 in \ the ring Z/aZ[x] with n natural? I´m working in some kind of proof of the best known theorem that says it´s impossible there exist a ...